| L(s) = 1 | − 2-s + 4-s + 2·5-s + 2·7-s − 8-s + 2·9-s − 2·10-s − 2·14-s + 16-s − 4·17-s − 2·18-s − 8·19-s + 2·20-s + 2·23-s + 2·25-s + 2·28-s − 10·31-s − 32-s + 4·34-s + 4·35-s + 2·36-s + 2·37-s + 8·38-s − 2·40-s + 8·41-s + 4·43-s + 4·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.755·7-s − 0.353·8-s + 2/3·9-s − 0.632·10-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.471·18-s − 1.83·19-s + 0.447·20-s + 0.417·23-s + 2/5·25-s + 0.377·28-s − 1.79·31-s − 0.176·32-s + 0.685·34-s + 0.676·35-s + 1/3·36-s + 0.328·37-s + 1.29·38-s − 0.316·40-s + 1.24·41-s + 0.609·43-s + 0.596·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8890266165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8890266165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 313 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T - 2 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.7018078077, −16.0604919641, −15.8500335295, −14.9610771670, −14.5985305972, −14.4689634218, −13.3918975116, −13.0632154356, −12.7554355820, −12.0068243288, −11.1621903372, −10.9314601100, −10.5044065723, −9.73330544301, −9.31547594898, −8.71039173430, −8.24441244586, −7.44297106705, −6.86304134930, −6.26907002430, −5.55005158695, −4.68310664056, −3.95671936707, −2.44266592155, −1.72755939483,
1.72755939483, 2.44266592155, 3.95671936707, 4.68310664056, 5.55005158695, 6.26907002430, 6.86304134930, 7.44297106705, 8.24441244586, 8.71039173430, 9.31547594898, 9.73330544301, 10.5044065723, 10.9314601100, 11.1621903372, 12.0068243288, 12.7554355820, 13.0632154356, 13.3918975116, 14.4689634218, 14.5985305972, 14.9610771670, 15.8500335295, 16.0604919641, 16.7018078077
